Calculus Examples

$\int_{0}^{\ln (2)} 2 e^{3 x} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{\ln (2)} e^{3 x} d x$

Step 2

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 2.1.4

Multiply $3$ by $1$ .

$3$

Step 2.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot 0$

Step 2.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 \ln (2)$

Step 2.5

Simplify.

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Step 2.5.1

Simplify $3 \ln (2)$ by moving $3$ inside the logarithm.

$u_{upper} = \ln (2^{3})$

Step 2.5.2

Raise $2$ to the power of $3$ .

$u_{upper} = \ln (8)$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \ln (8)$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$2 \int_{0}^{\ln (8)} e^{u} \frac{1}{3} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{3}$ .

$2 \int_{0}^{\ln (8)} \frac{e^{u}}{3} d u$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$2 (\frac{1}{3} \int_{0}^{\ln (8)} e^{u} d u)$

Step 5

Combine $\frac{1}{3}$ and $2$ .

$\frac{2}{3} \int_{0}^{\ln (8)} e^{u} d u$

Step 6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\frac{2}{3} e^{u}]_{0}^{\ln (8)}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $e^{u}$ at $\ln (8)$ and at $0$ .

$\frac{2}{3} ((e^{\ln (8)}) - e^{0})$

Step 7.2

Simplify.

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Step 7.2.1

Exponentiation and log are inverse functions.

$\frac{2}{3} (8 - e^{0})$

Step 7.2.2

Anything raised to $0$ is $1$ .

$\frac{2}{3} (8 - 1 \cdot 1)$

Step 7.2.3

Multiply $- 1$ by $1$ .

$\frac{2}{3} (8 - 1)$

Step 7.2.4

Subtract $1$ from $8$ .

$\frac{2}{3} \cdot 7$

Step 7.2.5

Combine $\frac{2}{3}$ and $7$ .

$\frac{2 \cdot 7}{3}$

Step 7.2.6

Multiply $2$ by $7$ .

$\frac{14}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{14}{3}$

Decimal Form:

$4.\overline{6}$

Mixed Number Form:

$4 \frac{2}{3}$

Step 9